The particle in the box model system is the simplest non-trivial application of the Schrödinger equation, but one which illustrates many of the fundamental concepts of quantum mechanics.
For a particle moving in one dimension (again along the x- axis), the Schrödinger equation can be written.
- ℏ is the reduced Planck Constant
- m is the mass of the particle
- ψ(x) is the stationary time-independent wavefunction
- V(x) is the potential energy as a function of position
- E is the energy, a real number
This equation can be modified for a particle of mass m free to move parallel to the x- axis with zero potential energy (V = 0 everywhere) resulting in the quantum mechanical description of free motion in one dimension:
Assume that the particle can move freely between two end points x = 0 and x = L, but cannot penetrate past either end. This is equivalent to a potential energy dependent on x.
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